The EU wants Northern Ireland to leave the UK and stay in the EU.

No it doesn't. It says the best way of avoiding a hard border is to work to the same rules on goods, but it is not proposing any change to the governance, or that NI should make payments to the EU, or that it must have free movement or anything else about the EU. It would not be part of the EU if the proposal was adopted.

It makes as much, if not more, sense for the Irish Republic to leave the EU and join the UK. Look where the trade goes

The Republic is a self governing independent country within the EU that will do what it thinks is in its own best interests. Having fought to gain that independence from the UK it is hardly likely to give it up to help Brexiteers out.